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The problem asks us to calculate the equilibrium angle, \(\theta\), formed by a uniform semicircular rod of radius \(r\), which is pinned at its top and allowed to oscillate freely in a vertical plane. 

### Steps to solve:

1. **Center of Mass Calculation:**
   - For a uniform semicircular rod, the center of mass is not at the geometric center, but slightly below. The vertical distance of the center of mass from the pin is given by:
     \[
     y_{CM} = \frac{2r}{\pi}
     \]
   - This is the distance from the pin \(O\) to the center of mass of the semicircular rod.

2. **Equilibrium Condition:**
   - At equilibrium, the potential energy is minimized. This happens when the center of mass is directly below the pivot. The angle \(\theta\) is defined as the angle between the rod’s diameter and the vertical direction at equilibrium.

3. **Geometrical Relations:**
   - From the geometry, the center of mass forms a right triangle with the pivot \(O\) and the lowest point of the rod. The height of the center of mass from the lowest point is \(r - \frac{2r}{\pi}\).
   - The equilibrium angle \(\theta\) is related to this geometry, and can be approximated using trigonometric relations, typically by solving:
     \[
     \theta = \cos^{-1}\left(\frac{y_{CM}}{r}\right)
     \]
   - Substituting \(y_{CM} = \frac{2r}{\pi}\):
     \[
     \theta = \cos^{-1}\left(\frac{2}{\pi}\right)
     \]

4. **Numerical Calculation:**
   - Now, calculating this expression:
     \[
     \theta = \cos^{-1}\left(\frac{2}{3.1416}\right) = \cos^{-1}(0.6366)
     \]

Let me calculate this to get the result.The calculated angle \(\theta\) is approximately \(50.46^\circ\). However, this value doesn't match any of the options provided. This suggests there may be additional factors or approximations in the problem that affect the angle calculation, such as the rod's shape or assumptions regarding equilibrium.

Would you like further clarification on the steps or a discussion on how to approach it differently?

Here are five related questions to deepen your understanding:

1. How does the center of mass of a uniform semicircular rod differ from a straight rod?
2. Why is the potential energy minimized at equilibrium?
3. What other methods could be used to solve for \(\theta\) if given different conditions?
4. How would the angle change if the rod had a different shape (e.g., full circular)?
5. What are the typical applications of these types of equilibrium problems in physics?

**Tip:** Always double-check if additional constraints or assumptions are provided in equilibrium problems, as they might affect the calculation.

Una varilla semicircular uniforme de radio r está apoyada en un pin en su extremo superior y puede oscilar libremente en el plano vertical. Calcula el ángulo θ que forma el diámetro con la vertical para la posición de equilibrio.

Calculate the equilibrium angle θ for a uniform semicircular rod of radius r, oscillating freely in the vertical plane. Learn the application of geometry, center of mass calculations, and trigonometric functions to determine the equilibrium condition.

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